# Theory Transfer

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theory Transfer
imports Hilbert_Choice
`(*  Title:      HOL/Transfer.thy    Author:     Brian Huffman, TU Muenchen*)header {* Generic theorem transfer using relations *}theory Transferimports Plain Hilbert_Choicebeginsubsection {* Relator for function space *}definition  fun_rel :: "('a => 'c => bool) => ('b => 'd => bool) => ('a => 'b) => ('c => 'd) => bool" (infixr "===>" 55)where  "fun_rel A B = (λf g. ∀x y. A x y --> B (f x) (g y))"lemma fun_relI [intro]:  assumes "!!x y. A x y ==> B (f x) (g y)"  shows "(A ===> B) f g"  using assms by (simp add: fun_rel_def)lemma fun_relD:  assumes "(A ===> B) f g" and "A x y"  shows "B (f x) (g y)"  using assms by (simp add: fun_rel_def)lemma fun_relD2:  assumes "(A ===> B) f g" and "A x x"  shows "B (f x) (g x)"  using assms unfolding fun_rel_def by autolemma fun_relE:  assumes "(A ===> B) f g" and "A x y"  obtains "B (f x) (g y)"  using assms by (simp add: fun_rel_def)lemma fun_rel_eq:  shows "((op =) ===> (op =)) = (op =)"  by (auto simp add: fun_eq_iff elim: fun_relE)lemma fun_rel_eq_rel:  shows "((op =) ===> R) = (λf g. ∀x. R (f x) (g x))"  by (simp add: fun_rel_def)subsection {* Transfer method *}text {* Explicit tag for relation membership allows for  backward proof methods. *}definition Rel :: "('a => 'b => bool) => 'a => 'b => bool"  where "Rel r ≡ r"text {* Handling of equality relations *}definition is_equality :: "('a => 'a => bool) => bool"  where "is_equality R <-> R = (op =)"text {* Handling of meta-logic connectives *}definition transfer_forall where  "transfer_forall ≡ All"definition transfer_implies where  "transfer_implies ≡ op -->"definition transfer_bforall :: "('a => bool) => ('a => bool) => bool"  where "transfer_bforall ≡ (λP Q. ∀x. P x --> Q x)"lemma transfer_forall_eq: "(!!x. P x) ≡ Trueprop (transfer_forall (λx. P x))"  unfolding atomize_all transfer_forall_def ..lemma transfer_implies_eq: "(A ==> B) ≡ Trueprop (transfer_implies A B)"  unfolding atomize_imp transfer_implies_def ..lemma transfer_bforall_unfold:  "Trueprop (transfer_bforall P (λx. Q x)) ≡ (!!x. P x ==> Q x)"  unfolding transfer_bforall_def atomize_imp atomize_all ..lemma transfer_start: "[|P; Rel (op =) P Q|] ==> Q"  unfolding Rel_def by simplemma transfer_start': "[|P; Rel (op -->) P Q|] ==> Q"  unfolding Rel_def by simplemma transfer_prover_start: "[|x = x'; Rel R x' y|] ==> Rel R x y"  by simplemma Rel_eq_refl: "Rel (op =) x x"  unfolding Rel_def ..lemma Rel_app:  assumes "Rel (A ===> B) f g" and "Rel A x y"  shows "Rel B (f x) (g y)"  using assms unfolding Rel_def fun_rel_def by fastlemma Rel_abs:  assumes "!!x y. Rel A x y ==> Rel B (f x) (g y)"  shows "Rel (A ===> B) (λx. f x) (λy. g y)"  using assms unfolding Rel_def fun_rel_def by fastML_file "Tools/transfer.ML"setup Transfer.setupdeclare refl [transfer_rule]declare fun_rel_eq [relator_eq]hide_const (open) Relsubsection {* Predicates on relations, i.e. ``class constraints'' *}definition right_total :: "('a => 'b => bool) => bool"  where "right_total R <-> (∀y. ∃x. R x y)"definition right_unique :: "('a => 'b => bool) => bool"  where "right_unique R <-> (∀x y z. R x y --> R x z --> y = z)"definition bi_total :: "('a => 'b => bool) => bool"  where "bi_total R <-> (∀x. ∃y. R x y) ∧ (∀y. ∃x. R x y)"definition bi_unique :: "('a => 'b => bool) => bool"  where "bi_unique R <->    (∀x y z. R x y --> R x z --> y = z) ∧    (∀x y z. R x z --> R y z --> x = y)"lemma right_total_alt_def:  "right_total R <-> ((R ===> op -->) ===> op -->) All All"  unfolding right_total_def fun_rel_def  apply (rule iffI, fast)  apply (rule allI)  apply (drule_tac x="λx. True" in spec)  apply (drule_tac x="λy. ∃x. R x y" in spec)  apply fast  donelemma right_unique_alt_def:  "right_unique R <-> (R ===> R ===> op -->) (op =) (op =)"  unfolding right_unique_def fun_rel_def by autolemma bi_total_alt_def:  "bi_total R <-> ((R ===> op =) ===> op =) All All"  unfolding bi_total_def fun_rel_def  apply (rule iffI, fast)  apply safe  apply (drule_tac x="λx. ∃y. R x y" in spec)  apply (drule_tac x="λy. True" in spec)  apply fast  apply (drule_tac x="λx. True" in spec)  apply (drule_tac x="λy. ∃x. R x y" in spec)  apply fast  donelemma bi_unique_alt_def:  "bi_unique R <-> (R ===> R ===> op =) (op =) (op =)"  unfolding bi_unique_def fun_rel_def by autotext {* Properties are preserved by relation composition. *}lemma OO_def: "R OO S = (λx z. ∃y. R x y ∧ S y z)"  by autolemma bi_total_OO: "[|bi_total A; bi_total B|] ==> bi_total (A OO B)"  unfolding bi_total_def OO_def by metislemma bi_unique_OO: "[|bi_unique A; bi_unique B|] ==> bi_unique (A OO B)"  unfolding bi_unique_def OO_def by metislemma right_total_OO:  "[|right_total A; right_total B|] ==> right_total (A OO B)"  unfolding right_total_def OO_def by metislemma right_unique_OO:  "[|right_unique A; right_unique B|] ==> right_unique (A OO B)"  unfolding right_unique_def OO_def by metissubsection {* Properties of relators *}lemma is_equality_eq [transfer_rule]: "is_equality (op =)"  unfolding is_equality_def by simplemma right_total_eq [transfer_rule]: "right_total (op =)"  unfolding right_total_def by simplemma right_unique_eq [transfer_rule]: "right_unique (op =)"  unfolding right_unique_def by simplemma bi_total_eq [transfer_rule]: "bi_total (op =)"  unfolding bi_total_def by simplemma bi_unique_eq [transfer_rule]: "bi_unique (op =)"  unfolding bi_unique_def by simplemma right_total_fun [transfer_rule]:  "[|right_unique A; right_total B|] ==> right_total (A ===> B)"  unfolding right_total_def fun_rel_def  apply (rule allI, rename_tac g)  apply (rule_tac x="λx. SOME z. B z (g (THE y. A x y))" in exI)  apply clarify  apply (subgoal_tac "(THE y. A x y) = y", simp)  apply (rule someI_ex)  apply (simp)  apply (rule the_equality)  apply assumption  apply (simp add: right_unique_def)  donelemma right_unique_fun [transfer_rule]:  "[|right_total A; right_unique B|] ==> right_unique (A ===> B)"  unfolding right_total_def right_unique_def fun_rel_def  by (clarify, rule ext, fast)lemma bi_total_fun [transfer_rule]:  "[|bi_unique A; bi_total B|] ==> bi_total (A ===> B)"  unfolding bi_total_def fun_rel_def  apply safe  apply (rename_tac f)  apply (rule_tac x="λy. SOME z. B (f (THE x. A x y)) z" in exI)  apply clarify  apply (subgoal_tac "(THE x. A x y) = x", simp)  apply (rule someI_ex)  apply (simp)  apply (rule the_equality)  apply assumption  apply (simp add: bi_unique_def)  apply (rename_tac g)  apply (rule_tac x="λx. SOME z. B z (g (THE y. A x y))" in exI)  apply clarify  apply (subgoal_tac "(THE y. A x y) = y", simp)  apply (rule someI_ex)  apply (simp)  apply (rule the_equality)  apply assumption  apply (simp add: bi_unique_def)  donelemma bi_unique_fun [transfer_rule]:  "[|bi_total A; bi_unique B|] ==> bi_unique (A ===> B)"  unfolding bi_total_def bi_unique_def fun_rel_def fun_eq_iff  by (safe, metis, fast)subsection {* Transfer rules *}text {* Transfer rules using implication instead of equality on booleans. *}lemma eq_imp_transfer [transfer_rule]:  "right_unique A ==> (A ===> A ===> op -->) (op =) (op =)"  unfolding right_unique_alt_def .lemma forall_imp_transfer [transfer_rule]:  "right_total A ==> ((A ===> op -->) ===> op -->) transfer_forall transfer_forall"  unfolding right_total_alt_def transfer_forall_def .lemma eq_transfer [transfer_rule]:  assumes "bi_unique A"  shows "(A ===> A ===> op =) (op =) (op =)"  using assms unfolding bi_unique_def fun_rel_def by autolemma All_transfer [transfer_rule]:  assumes "bi_total A"  shows "((A ===> op =) ===> op =) All All"  using assms unfolding bi_total_def fun_rel_def by fastlemma Ex_transfer [transfer_rule]:  assumes "bi_total A"  shows "((A ===> op =) ===> op =) Ex Ex"  using assms unfolding bi_total_def fun_rel_def by fastlemma If_transfer [transfer_rule]: "(op = ===> A ===> A ===> A) If If"  unfolding fun_rel_def by simplemma Let_transfer [transfer_rule]: "(A ===> (A ===> B) ===> B) Let Let"  unfolding fun_rel_def by simplemma id_transfer [transfer_rule]: "(A ===> A) id id"  unfolding fun_rel_def by simplemma comp_transfer [transfer_rule]:  "((B ===> C) ===> (A ===> B) ===> (A ===> C)) (op o) (op o)"  unfolding fun_rel_def by simplemma fun_upd_transfer [transfer_rule]:  assumes [transfer_rule]: "bi_unique A"  shows "((A ===> B) ===> A ===> B ===> A ===> B) fun_upd fun_upd"  unfolding fun_upd_def [abs_def] by transfer_proverlemma nat_case_transfer [transfer_rule]:  "(A ===> (op = ===> A) ===> op = ===> A) nat_case nat_case"  unfolding fun_rel_def by (simp split: nat.split)lemma nat_rec_transfer [transfer_rule]:  "(A ===> (op = ===> A ===> A) ===> op = ===> A) nat_rec nat_rec"  unfolding fun_rel_def by (clarsimp, rename_tac n, induct_tac n, simp_all)lemma funpow_transfer [transfer_rule]:  "(op = ===> (A ===> A) ===> (A ===> A)) compow compow"  unfolding funpow_def by transfer_provertext {* Fallback rule for transferring universal quantifiers over  correspondence relations that are not bi-total, and do not have  custom transfer rules (e.g. relations between function types). *}lemma Domainp_iff: "Domainp T x <-> (∃y. T x y)"  by autolemma Domainp_forall_transfer [transfer_rule]:  assumes "right_total A"  shows "((A ===> op =) ===> op =)    (transfer_bforall (Domainp A)) transfer_forall"  using assms unfolding right_total_def  unfolding transfer_forall_def transfer_bforall_def fun_rel_def Domainp_iff  by metistext {* Preferred rule for transferring universal quantifiers over  bi-total correspondence relations (later rules are tried first). *}lemma forall_transfer [transfer_rule]:  "bi_total A ==> ((A ===> op =) ===> op =) transfer_forall transfer_forall"  unfolding transfer_forall_def by (rule All_transfer)end`